> Isn't the point to train your brain to think in a certain way
You might get that through other methods besides trig, meaning trig isn't strictly necessary -- it's just one option among potentially many. The hard part is proving efficacy and efficiency of each option.
CompSci interview questions go way beyond loops, conditionals and recursion though, such that if you don't already know some secret sauces, it's unlikely to get the best case solution that the interviewer wants.
The other thing throwing this off is interviews are looking for different grades of solutions or maybe just looking at your problem solving/communication skills, e.g. you can fail to find the best known solution but still get a job because of your reasoning.
So in this way, the algorithms & data structures interviews can sometimes be used to test knowledge, or test problem solving ability, or both.
The interview stopped being a test of "if you can do this job" and instead seeks to find a correlation backed up by some research that says "people that prepare, study, and do well at this interview also do well at this company". Similar research meant that they needed to stop asking brain teaser questions because it wasn't a good indicator of success.